An Almost Periodic Function of Several Variables with no Local Minimum

An almost periodic function is a generalization of a periodic function. Almost periodic functions whose domain is the real numbers are well understood. The notion of almost periodic is easily generalized to higher-dimensional Euclidean space and even other topological groups. The properties of almost periodic functions on such other domains appear to be less well known. This paper proves that these properties may be very different. Specifically, an almost periodic function of two or more real variables need not have a local minimum (unlike an a.p. function of one variable).